Problem: $\sum\limits_{n=1}^{\infty}e^{-n}$ Does the integral test apply to the series? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Explanation: $e^{-x}$ is continuous and positive for all $x\geq 1$. To find whether it's always decreasing for $x\geq1$, let's consider its derivative. $\dfrac{d}{dx}\left(e^{-x}\right)= -\dfrac{1}{e^x}$ Because $e^x$ is always positive, the expression $ -\dfrac{1}{e^x}$ is always negative. So $\dfrac{d}{dx}\left(\dfrac{1}{e^x}\right)$ is negative for all $x\geq 1$, which means $\dfrac{1}{e^x}$ is decreasing. In conclusion, the integral test does apply to the series.